Problem: Simplify; express your answer in exponential form. Assume $x\neq 0, k\neq 0$. $\dfrac{{(x^{-1}k^{-3})^{3}}}{{(x^{5}k^{2})^{-5}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(x^{-1}k^{-3})^{3} = (x^{-1})^{3}(k^{-3})^{3}}$ On the left, we have ${x^{-1}}$ to the exponent ${3}$ . Now ${-1 \times 3 = -3}$ , so ${(x^{-1})^{3} = x^{-3}}$ Apply the ideas above to simplify the equation. $\dfrac{{(x^{-1}k^{-3})^{3}}}{{(x^{5}k^{2})^{-5}}} = \dfrac{{x^{-3}k^{-9}}}{{x^{-25}k^{-10}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{-3}k^{-9}}}{{x^{-25}k^{-10}}} = \dfrac{{x^{-3}}}{{x^{-25}}} \cdot \dfrac{{k^{-9}}}{{k^{-10}}} = x^{{-3} - {(-25)}} \cdot k^{{-9} - {(-10)}} = x^{22}k$